Textbook Question
Determine the following limits.
lim x→∞ x^4+7 / x^5+x^2−x
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1. Limits and Continuity
Finding Limits Algebraically
2:19 minutesProblem 2.35
Textbook Question
Determine the following limits.
lim x→∞ sin x / e^x
Verified step by step guidance1
Step 1: Understand the problem. We need to find the limit of the function \( \frac{\sin x}{e^x} \) as \( x \) approaches infinity.
Step 2: Analyze the behavior of the numerator and the denominator separately. The function \( \sin x \) oscillates between -1 and 1 for all \( x \).
Step 3: Consider the behavior of the denominator \( e^x \). As \( x \) approaches infinity, \( e^x \) grows exponentially and becomes very large.
Step 4: Apply the Squeeze Theorem. Since \( -1 \leq \sin x \leq 1 \), we have \( -\frac{1}{e^x} \leq \frac{\sin x}{e^x} \leq \frac{1}{e^x} \).
Step 5: Evaluate the limits of the bounding functions. As \( x \to \infty \), both \( \frac{1}{e^x} \to 0 \) and \( -\frac{1}{e^x} \to 0 \). By the Squeeze Theorem, \( \lim_{x \to \infty} \frac{\sin x}{e^x} = 0 \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. In this context, we analyze how the function behaves as x becomes very large, which can reveal whether the function approaches a specific value, diverges, or oscillates.
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One-Sided Limits
Behavior of Sinusoidal Functions
The sine function oscillates between -1 and 1 for all real numbers. This bounded behavior is crucial when evaluating limits involving sine, as it indicates that despite the oscillation, the overall contribution of sin(x) becomes negligible compared to other functions that grow without bound, such as exponential functions.
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5:46Graphs of Exponential Functions
Exponential Growth
Exponential functions, like e^x, grow significantly faster than polynomial or sinusoidal functions as x approaches infinity. This rapid growth is key in limit problems, as it often leads to the conclusion that terms involving e^x will dominate the behavior of the limit, driving the overall limit towards zero when combined with bounded functions.
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